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I'm not sure I quite understand why systems with fractal systems show power-law behavior. My "gut" understanding is that the power-law index indicates the correct scaling factor for the system so that the original and scaled systems look "similar", and is indicaative of the fractal dimension of the system. I'm still having difficulty seeing this physically; is there a good physical explanation of why fractal systems show power-law scaling? For example, in the famous coastline of Britain measurement, why does decreasing the length scale of my 'ruler' increase length as a power-law, and not say exponential, or some other crazy function? What is so special or universal about the power-law behavior?

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It's because fractal systems are, pretty much by definition, self-similar which means that there is no preferred length scale. If something else depends on the length scale $L$ as a function $f(L)$, the argument $L$ must have units – and no unit is better than any other – so it is "dimensionful". On the other hand, $f(L)$ is a quantity that must have well-defined units, too, because there can't be any preferred value, either. A preferred value of $f(L)$ would translate to a preferred value of $L$, and so on.

The powers $A\cdot L^\kappa$ (with a dimensionless $A$) are the only functions with well-defined units that don't require any dimensionful parameter – i.e. effectively preferred length scale – to be defined. This contrasts with things like $\exp(L/L_0)$. The argument of the exponential must be dimensionless so $L$ would have to be divided by $L_0$, a preferred length scale, for it to be allowed in the exponent. But that would violate the self-similarity because patterns at distance scale $L_0$ would be closer to "fundamental rules of the fractal" than all other distance scales, both longer and shorter.

The critical exponents are important in critical systems and conformal field theories in physics – in this respect, they're analogous to fractals.

You may also define fractals independently of self-similarity, via their fractal dimension. Then your question is answered because it follows almost from the definition of the fractal dimension where the number of balls $N\sim \epsilon^{-D}$ so the power law is incorporated into the definition of the fractal dimension and it propagates from $N$ to any other "derived" quantity you may consider.

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